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Question: The resistance of a wire of length 300m and cross-sectional area \(1.0m{{m}^{2}}\) made of material ...

The resistance of a wire of length 300m and cross-sectional area 1.0mm21.0m{{m}^{2}} made of material of resistivity 1.0×10−7Ωm1.0\times {{10}^{-7}}\Omega m is:
(A)2Ω (B)3Ω (C)20Ω (D)30Ω \begin{aligned} & (A)2\Omega \\\ & (B)3\Omega \\\ & (C)20\Omega \\\ & (D)30\Omega \\\ \end{aligned}

Explanation

Solution

We have to calculate the total resistance of the wire for the given values of different parameters. We shall use the formula of resistance, which is, the resistance of a wire can be written as the product of its length and resistivity upon its cross-sectional area. Also, before putting all the values in the mentioned formula, we will make sure they all have the same units.

Complete answer:
Let the length of the wire be given by LL.
Then it has been given to us in the problem that the value of LL is 300 meters.
Now, let the area of cross-section of the wire be A. Then, it has been given to us that:
⇒A=1.0mm2\Rightarrow A=1.0m{{m}^{2}}
We will convert this Area of cross-section into meter squared so that all the units remain consistent with each other upon calculation:
Since,
⇒1mm=10−3m\Rightarrow 1mm={{10}^{-3}}m
Therefore,
⇒1mm2=10−6m2\Rightarrow 1m{{m}^{2}}={{10}^{-6}}{{m}^{2}}
And, let the resistivity of the given wire be denoted by ρ\rho .
Also, the value of resistivity has been given to us in the problem as:
⇒ρ=1.0×10−7Ωm\Rightarrow \rho =1.0\times {{10}^{-7}}\Omega m
Now, we can calculate the resistance of the metal wire with the help of the given formula:
⇒R=ρLA\Rightarrow R=\dfrac{\rho L}{A}
Putting the values of all the terms in Right Hand Side of the equation, we get:
⇒R=1.0×10−7×30010−6 ⇒R=300×10−1 ⇒R=30Ω \begin{aligned} & \Rightarrow R=\dfrac{1.0\times {{10}^{-7}}\times 300}{{{10}^{-6}}} \\\ & \Rightarrow R=300\times {{10}^{-1}} \\\ & \Rightarrow R=30\Omega \\\ \end{aligned}
Hence, the resistance of the wire comes out to be 30Ω30\Omega .

Hence, option (D) is the correct option.

Note:
These are some basic formulas from one of the most important chapters of Physics, Current Electricity. So, one should very well be aware of these basic formulas. Also, one should always keep check of their calculations at every step of solution or else one small error might end up giving a wrong solution to our problem.